CauchySchwarzBunyakovsky inequality

Today I learned from Introduction to Metric and Topological Spaces by W.A. Sutherland a version of the CauchySchwarz inequality involving integrals. It goes by the funny name of CauchySchwarzBunyakovsky inequality.

First, lets recall the CauchySchwarz inequality. It states

for al real numbers

.

Proof:

The inequality is obviously true if

. Hence we may assume that at least one is not 0.

Then

Treating the LHS as a quadratic in

, we see that its discriminant cannot be positive.

This is the proof given in Introduction to Metric and Topological Spaces.

The integral version is as follows.

The proof is similar to the CauchySchwarz case, only this time you start with

Re: CauchySchwarzBunyakovsky inequality

That's nice. I remember we have a similar proof in Calculus for multi-variable function's Taylor expansion. It adds in lamda as well. Such method is called "adding parameter", which shares the same delicacy as adding a line to solve geometry problems.

Re: CauchySchwarzBunyakovsky inequality

Basically, the roots cannot be real. Consider the quadratic equation as a parabola. If the equation has real roots then it crosses the x axis twice and has negative values between them, but we know that our quadratic function cannot be negative, so the roots have to be imaginary and so the discriminant has to be less than zero.